In any event, the real test is when you turn the ignition on, what does the THEFT light do? If it flashes, don't waste battery power seeing if it will start.
With the THEFT light flashing occasionally in my 98 Mountaineer for the first year I had it, I finally learned that my random issue could be predicted by the theft light. Once I knew that the flashing theft light meant the engine would not start, I turned the ignition off, and then turned it back on. It would eventually not flash, and the engine would start and I could drive away. It happened rarely, maybe 2-5 times a month. After I read about the components of the system and what others had done, discovered, I removed the plastic cover around the column and key. The transponder was not attached very well.
I'm not saying that's your issue, but that you need to watch the THEFT light. Do not try to start the engine unless that light stays on and does not flash with the ignition on. Once that happens, it may start if the battery is charged and nothing else is wrong with the engine etc. That's all the PATS system does, it verifies that a proper key code has been received prior to the engine starting.